Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 6.0 → 5.1

Time: 8.9s

Precision: 64

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.26584703889505767 \cdot 10^{182}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(1 \cdot {\left({x}^{\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{3}}\right)}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - 0.0027777777777778 \cdot \frac{z}{x}\right) + \frac{0.0833333333333329956}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r642370 = x;
        double r642371 = 0.5;
        double r642372 = r642370 - r642371;
        double r642373 = log(r642370);
        double r642374 = r642372 * r642373;
        double r642375 = r642374 - r642370;
        double r642376 = 0.91893853320467;
        double r642377 = r642375 + r642376;
        double r642378 = y;
        double r642379 = 0.0007936500793651;
        double r642380 = r642378 + r642379;
        double r642381 = z;
        double r642382 = r642380 * r642381;
        double r642383 = 0.0027777777777778;
        double r642384 = r642382 - r642383;
        double r642385 = r642384 * r642381;
        double r642386 = 0.083333333333333;
        double r642387 = r642385 + r642386;
        double r642388 = r642387 / r642370;
        double r642389 = r642377 + r642388;
        return r642389;
}

↓

double f(double x, double y, double z) {
        double r642390 = x;
        double r642391 = 1.2658470388950577e+182;
        bool r642392 = r642390 <= r642391;
        double r642393 = 0.5;
        double r642394 = r642390 - r642393;
        double r642395 = cbrt(r642390);
        double r642396 = r642395 * r642395;
        double r642397 = log(r642396);
        double r642398 = r642394 * r642397;
        double r642399 = 1.0;
        double r642400 = 0.3333333333333333;
        double r642401 = cbrt(r642400);
        double r642402 = r642401 * r642401;
        double r642403 = pow(r642390, r642402);
        double r642404 = pow(r642403, r642401);
        double r642405 = r642399 * r642404;
        double r642406 = log(r642405);
        double r642407 = r642406 * r642394;
        double r642408 = r642407 - r642390;
        double r642409 = r642398 + r642408;
        double r642410 = 0.91893853320467;
        double r642411 = r642409 + r642410;
        double r642412 = y;
        double r642413 = 0.0007936500793651;
        double r642414 = r642412 + r642413;
        double r642415 = z;
        double r642416 = r642414 * r642415;
        double r642417 = 0.0027777777777778;
        double r642418 = r642416 - r642417;
        double r642419 = r642418 * r642415;
        double r642420 = 0.083333333333333;
        double r642421 = r642419 + r642420;
        double r642422 = r642421 / r642390;
        double r642423 = r642411 + r642422;
        double r642424 = log(r642395);
        double r642425 = r642424 * r642394;
        double r642426 = r642425 - r642390;
        double r642427 = r642398 + r642426;
        double r642428 = r642427 + r642410;
        double r642429 = 2.0;
        double r642430 = pow(r642415, r642429);
        double r642431 = r642430 / r642390;
        double r642432 = r642413 * r642431;
        double r642433 = r642415 / r642390;
        double r642434 = r642417 * r642433;
        double r642435 = r642432 - r642434;
        double r642436 = r642420 / r642390;
        double r642437 = r642435 + r642436;
        double r642438 = r642428 + r642437;
        double r642439 = r642392 ? r642423 : r642438;
        return r642439;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.3
Herbie	5.1

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

1. Split input into 2 regimes

2. if x < 1.2658470388950577e+182

   1. Initial program 2.9

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 2.9

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   4. Applied log-prod 3.0

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   5. Applied distribute-lft-in 3.0

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   6. Applied associate--l+ 3.0

      \[\leadsto \left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right)\right)} + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   7. Simplified 3.0

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \color{blue}{\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)}\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 3.0

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{\color{blue}{1 \cdot x}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   10. Applied cbrt-prod 3.0

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{x}\right)} \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   11. Simplified 3.0

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\color{blue}{1} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   12. Simplified 2.9

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(1 \cdot \color{blue}{{x}^{\frac{1}{3}}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 3.0

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(1 \cdot {x}^{\color{blue}{\left(\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right) \cdot \sqrt[3]{\frac{1}{3}}\right)}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   15. Applied pow-unpow 3.0

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(1 \cdot \color{blue}{{\left({x}^{\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{3}}\right)}}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   if 1.2658470388950577e+182 < x

   1. Initial program 14.2

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 14.2

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   4. Applied log-prod 14.2

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   5. Applied distribute-lft-in 14.3

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   6. Applied associate--l+ 14.2

      \[\leadsto \left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right)\right)} + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   7. Simplified 14.2

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \color{blue}{\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)}\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   8. Taylor expanded around 0 10.7

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \color{blue}{\left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]

   9. Simplified 10.7

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \color{blue}{\left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - 0.0027777777777778 \cdot \frac{z}{x}\right) + \frac{0.0833333333333329956}{x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.26584703889505767 \cdot 10^{182}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(1 \cdot {\left({x}^{\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{3}}\right)}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - 0.0027777777777778 \cdot \frac{z}{x}\right) + \frac{0.0833333333333329956}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))